Problem 62 - College Algebra

Question

$$\begin{aligned}&\text , \text { Perform the indicated operations. Indicate}\\\ &\text { the degree of the resulting polynomial.}\end{aligned}$$ $$\left(7 x^{4} y^{2}-5 x^{2} y^{2}+3 x y\right)+\left(-18 x^{4} y^{2}-6 x^{2} y^{2}-x y\right)$$

Step-by-Step Solution

Answer

The resulting polynomial after performing the operation is \(-11x^{4}y^{2} - 11x^{2}y^{2} + 2xy\) and its degree is 6.

Step 1: Combining Like Terms

When adding polynomials, we will combine 'like terms'. Like terms are those which have the same variables and powers. For the given polynomials, we will separately add the coefficients of \(x^{4}y^{2}\), \(x^{2}y^{2}\), and \(xy\):For term \(x^{4}y^{2}\): \(7 - 18 = -11\)For term \(x^{2}y^{2}\): \(-5 - 6 = -11\)For term \(xy\): \(3 - 1 = 2\)So, the resulting polynomial after adding these two is \(-11x^{4}y^{2} - 11x^{2}y^{2} + 2xy\).

Step 2: Identifying Higher Degree

The degree of a polynomial is defined as the highest degree of any term in the polynomial. In our resulting polynomial, the term with the highest degree is \(-11x^{4}y^{2}\) with degree 6 (considering both \(x^{4}\) and \(y^{2}\)).

Key Concepts

Combining Like Terms

Understanding how to master the task of combining like terms is essential in simplifying algebraic expressions, notably when dealing with polynomials. When adding or subtracting polynomials, you must align terms that have the same set of variables raised to the same powers—these are your 'like terms'. To add like terms effectively, you only add the coefficients (the numerical parts) while keeping the variable part unchanged.

For example, when you have polyomials with terms like \(7x^{4}y^{2}\) and \(-18x^{4}y^{2}\), you focus solely on the coefficients 7 and -18. When you combine them, you get \(7 - 18 = -11\), and the resulting term will be \(-11x^{4}y^{2}\). It's important to be methodical and cautious to ensure you're only combining coefficients of like terms to avoid errors.

Polynomial Degree

The concept of polynomial degree might appear daunting at first, but it is quite straightforward once you get the hang of it. The degree of a polynomial is the highest sum of the exponents of the variables in a single term within the polynomial. When you look at a term like \(x^{4}y^{2}\), for instance, the exponents of both x and y are added together to give you a degree of 6 for that term (since 4 + 2 = 6).Identifying the highest degree term in a polynomial tells you its overall degree. This is particularly important when classifying polynomials and in understanding their behavior especially in the context of graphing and solving polynomial functions. In the case of our previous example, our resulting polynomial after combining like terms would be \(-11x^{4}y^{2} - 11x^{2}y^{2} + 2xy\), which has a maximum degree of 6.

Algebraic Operations

Algebraic operations are the foundation of nearly all algebraic procedures, and they are part of everyday mathematical practices. These include addition, subtraction, multiplication, and division as applied to numbers, but they also extend to variables and exponents in algebra. When we perform operations with polynomials, we apply these same basic principles but need to be attentive to the rules of exponentiation and the distributive property.

In our exercise, addition is the primary operation executed. Even though it may seem simple at first glance, careful attention to detail is necessary: maintaining the integrity of variable terms, ensuring that only like terms are combined, and keeping an organized approach to manage multiple terms. Remember that while adding or subtracting unlike terms is not permissible, multiplying or dividing terms across a polynomial is possible, as long as the correct rules for exponentiation are followed.